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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreldmlmhm 16101 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom

Theoremlmimfn 16102 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso

Theoremislmhm 16103* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
Scalar       Scalar                                   LMHom

Theoremislmhm3 16104* Property of a module homomorphism, similar to ismhm 14740. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       Scalar                                   LMHom

Theoremlmhmlem 16105 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar       Scalar       LMHom

Theoremlmhmsca 16106 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar       Scalar       LMHom

Theoremlmghm 16107 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod2 16108 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod1 16109 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmf 16110 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlin 16111 A homomorphism of left modules is -linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar                                   LMHom

Theoremlmodvsinv 16112 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsinv2 16113 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar

Theoremislmhm2 16114* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 16009. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar       Scalar                                          LMHom

Theoremislmhmd 16115* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Scalar       Scalar                                                 LMHom

Theorem0lmhm 16116 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar       Scalar       LMHom

Theoremidlmhm 16117 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom

Theoreminvlmhm 16118 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom

Theoremlmhmco 16119 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom LMHom LMHom

Theoremlmhmplusg 16120 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom LMHom LMHom

Theoremlmhmvsca 16121 The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
Scalar              LMHom LMHom

Theoremlmhmf1o 16122 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMHom LMHom

Theoremlmhmima 16123 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmpreima 16124 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlsp 16125 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmrnlss 16126 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmkerlss 16127 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremreslmhm 16128 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LMHom LMHom

Theoremreslmhm2 16129 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
s               LMHom LMHom

Theoremreslmhm2b 16130 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
s               LMHom LMHom

Theoremlmhmeql 16131 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
LMHom LMHom

Theoremlspextmo 16132* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
LMHom

Theorempwsdiaglmhm 16133* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s                      LMHom

Theoremislmim 16134 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimf1o 16135 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso

Theoremlmimlmhm 16136 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimgim 16137 An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
LMIso GrpIso

Theoremislmim2 16138 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
LMIso LMHom LMHom

Theoremlmimcnv 16139 The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
LMIso LMIso

Theorembrlmic 16140 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LMIso

Theorembrlmici 16141 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMIso 𝑚

Theoremlmiclcl 16142 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚

Theoremlmicrcl 16143 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
𝑚

Theoremlmicsym 16144 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝑚 𝑚

Theoremlmhmpropd 16145* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar       Scalar       Scalar       Scalar                                                 LMHom LMHom

10.6.4  Subspace sum; bases for a left module

Syntaxclbs 16146 Extend class notation with the set of bases for a vector space.
LBasis

Definitiondf-lbs 16147* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis Scalar

Theoremislbs 16148* The predicate " is a basis for the left module or vector space ". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
Scalar                     LBasis

Theoremlbsss 16149 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsel 16150 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbssp 16151 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsind 16152 A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
LBasis              Scalar

Theoremlbsind2 16153 A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
LBasis              Scalar

Theoremlbspss 16154 No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis              Scalar

Theoremlsmcl 16155 The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Theoremlsmspsn 16156* Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
Scalar

Theoremlsmelval2 16157* Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)

Theoremlsmsp 16158 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlsmsp2 16159 Subspace sum of spans of subsets is the span of their union. (spanuni 23046 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlsmssspx 16160 Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)

Theoremlsmpr 16161 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)

Theoremlsppreli 16162 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlsmelpr 16163 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)

Theoremlsppr0 16164 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)

Theoremlsppr 16165* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
Scalar

Theoremlspprel 16166* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
Scalar

Theoremlspprabs 16167 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Theoremlspvadd 16168 The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntri 16169 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntrim 16170 Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlbspropd 16171* If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Scalar       Scalar                                          LBasis LBasis

Theorempj1lmhm 16172 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom

Theorempj1lmhm2 16173 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom s

10.7  Vector spaces

10.7.1  Definition and basic properties

Syntaxclvec 16174 Extend class notation with class of all left vector spaces.

Definitiondf-lvec 16175 Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremislvec 16176 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremlvecdrng 16177 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
Scalar

Theoremlveclmod 16178 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)

Theoremlsslvec 16179 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
s

Theoremlvecvs0or 16180 If a scalar product is zero, one of its factors must be zero. (hvmul0or 22527 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvsn0 16181 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
Scalar

Theoremlssvs0or 16182 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs. (Contributed by NM, 5-Apr-2015.)
Scalar

Theoremlvecvscan 16183 Cancellation law for scalar multiplication. (hvmulcan 22574 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvscan2 16184 Cancellation law for scalar multiplication. (hvmulcan2 22575 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecinv 16185 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
Scalar

Theoremlspsnvs 16186 A non-zero scalar product does not change the span of a singleton. (spansncol 23070 analog.) (Contributed by NM, 23-Apr-2014.)
Scalar

Theoremlspsneleq 16187 Membership relation that implies equality of spans. (spansneleq 23072 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspsncmp 16188 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)

Theoremlspsnne1 16189 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

Theoremlspsnne2 16190 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)

Theoremlspsnnecom 16191 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)

Theoremlspabs2 16192 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspabs3 16193 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspsneq 16194* Equal spans of singletons must have proportional vectors. See lspsnss2 16081 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Scalar

Theoremlspsneu 16195* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Scalar

Theoremlspsnel4 16196 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 23075 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspdisj 16197 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Theoremlspdisjb 16198 A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)

Theoremlspdisj2 16199 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)

Theoremlspfixed 16200* Show membership in the span of the sum of two vectors, one of which () is fixed in advance. (Contributed by NM, 27-May-2015.)

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